Private inductive types July 2013 Introduction Higher Inductive - - PowerPoint PPT Presentation

Private inductive types

July 2013

Introduction

◮ Higher Inductive types: adding equalities ◮ Preventing inconsistencies ◮ Preserving convertibility ◮ Simulating with private types

What is this thing called Equality

◮ A family of equality types: for x y : A, x = y is a type ◮ Described as an inductive type: no specific treatment ◮ Induction principle illuminating

∀A : Type.∀x : A. ∀P : A → Prop.P(x) ⇒ ∀y : A. x = y ⇒ P(y)

◮ If x = y then every property satisfied by x is also satisfied by y ◮ x and y are undistinguishable

◮ Are they really?

using a magnifying glass

◮ Say that when x = y, then x and y are not really the same

for all purposes

◮ So x = y should only mean there is a path between x and y ◮ Distinction at a microscopic level ◮ But at the macroscopic level, still x and y are equal.

Build new objects with paths between them

◮ State at the same time the creation of objects and the

property that they are identical.

◮ Example: assert the existence of two points N and S and two

paths between them.

◮ Already done easily for points using inductive types ◮ What about the paths?

◮ Natural to add paths as axioms

Inconsistencies with axiomatic paths

◮ Usual interpretation of equality (identity) types

◮ Ultimately only one way to build proofs of equality: reflexivity

◮ No confusion property of inductive types

◮ Rely on strong elimination

◮ Axiomatic paths between constructors incompatible with

no-confusion

Illustration

Inductive cellc := N | S. Axiom west : N = S. Axiom east : N = S.

◮ Obviously inconsistent in plain Coq.

Preventing inconsistency

◮ Allow only to define function that preserve path consistency ◮ In illustration, f N and f S must have a path between them. ◮ Also take into account dependent types ◮ Solution already easy to implement in Agda

Heavy solution

◮ Avoid inductive types ◮ State axioms for all elements of the higher inductive type

Illustrating the heavy solution

Parameters (cellc : Type) (N S : cellc). Axioms west east : N = S. Parameter cellc_rect (P : cellc -> Type) (vn : P N) (vs : P S) (pw : eq_rect N P vn S west = vs) (pe : eq_rect N P vn S east = vs) (x : cellc) : P x. Axiom cellc_rect_N := forall P vn vs pw pe, cellc_rect P vn vs pw pe N = vn. Axiom cellc_rect_S := forall P vn vs pw pe, cellc_rect P vn vs pw pe S = vs.

What’s wrong with being heavy?

◮ Provably equal is not convertible

◮ cellc rect P vn vs pw pe N and vn are not convertible

◮ More uses of eq rect are required everywhere ◮ The size of proofs increases drastically

Adding convertibility

◮ Come back to inductive types ◮ Design elimination function to enforce guarantees

Definition cellc_rect (P : cellc -> Type) (vn : P N) (vs : P S) (pw : eq_rect N P vn S west = vs) (pe : eq_rect N P vn S east = vs) (x : cellc) := match x return P x with N => vn | S => vs end.

Computing with cellc rect

◮ cellc rect P vn vs pw pe N and vn are now convertible ◮ Okay if the only functions definable in Coq have to be defined

using cellc rect.

◮ Need to forbid direct use of pattern-matching, tactics case,

discriminate, inversion, injection. . .

Idea of private types

◮ In a module, define an inductive type to be private ◮ Inside module: unsafe operations, trusting the programmer ◮ Outside module: more safety, only functions provided by

module designer

◮ Preserve computation (convertibility) for functions provided in

the module

◮ No modification of the kernel, only module handling ◮ Deactivate tactics and syntax ◮ Hard questions about consistency: not treated by the kernel

Simulating the circle inductive type

Module Circle. Local Inductive Circle := N | S. Axiom east : N = S. Axiom west : N = S. Definition circle_induction (A : Type)(vn : A)(vs : A) (epd : vn = vs)(wpd : vn = vs)(x : circle) : A := match x with N => vn | S => vs end. Axiom circle_induction_cws : forall A vn vs epd wpd, ap (circle_induction vn vs epd wpd) east_side = epd. End Circle.

Conclusion

◮ Potential inconsistency comes from adding axioms ◮ Idea of private types orthogonal to axioms ◮ Application outside homotopy theory are probable